$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$
-$$\lim _ { x \rightarrow \theta } \frac { \sin 4 x } { \sin 5 x }$$
A) 0
B) does not exist
C) $\frac { 5 } { 4 }$
D) $\frac { 4 } { 5 }$

Question

Quizplus · Accepted Answer

Using the limit property $\lim _ { x \rightarrow 0 } \frac { \sin x } { x } = 1$, we can manipulate the given limit to fit this form. By multiplying and dividing the expression by 4 and 5 respectively, we get $\lim _ { x \rightarrow \theta } \frac { 4 \sin 4 x } { 4 x } \cdot \frac { 5 x } { 5 \sin 5 x }$. As $x \rightarrow 0$, both $\frac { \sin 4 x } { 4 x }$ and $\frac { \sin 5 x } { 5 x }$ approach 1, leading to the result $\frac { 4 } { 5 }$.

Find the limit using lim⁡x=0sin⁡xx=1. \text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. } Find the limit using limx=0​xsinx​=1.

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$